Interpolation of Functions

On the divergence of trigonometric lacunary interpolation

“…Moreover, if ni < rj, then 5ni <= rj (cf. (5) By virtue of the estimate IIR~II < 5n (see [8, p. 246]) and (6,8,9) it follows at last that This implies (11), provided that both q > 0 is small and No (and thus rj) is large enough. To establish (12) one may proceed analogously.…”

“…[2; 4] and the literature cited there) one successively constructs a strictly n co increasing subsequence ( J)j=l with nl > No satisfying (5) njE{bk:kEN}, where No E N and 0 < q < 1 are to be chosen later. Note that (6) implies From (8,9) it follows that If t > 1/nl, then by (1,8,9) :-hq)…”

Fractal Interpolation on a Torus

“…, 2m, u j (ϕ k ) = δ kj . The procedure presented is very general and includes trigonometric functions (see for instance [8]), periodic splines, etc. Let us define the set…”

Mean convergence of hermite interpolation revisited

“…Sometimes we write w = w (a'~) to emphasize the parameters a and/3. We call w a generalized Jacobi weight function (w E G J), if it can be written as We should mention that our pn(w) is different from P(~'Z) in books such as [8] and [10] by a normalizing constant of order exactly v/ft. It is well known that p,~(w) has n distinct zeros in (-1,1).…”